Yes you can say: write this down or type this up

Although people will understand what you mean, this is not a very common or natural-sounding phrase. You can easily check questions like these for yourself by googling the full phrase in quotes. "Type down" returns a relatively small number of results, almost none of which have the meaning you intend (i.e., write down, but with a keyboard). If a phrase is common or natural it will yield tens of millions of google hits.